Function Spaces and Partial Differential Equations

My principal aim in this book is to present a comprehensive treatment of aspects of classical and modern analysis pertaining to Partial Differential Equations (PDEs) and their associated Function Spaces. I start with a quick review of the basic properties of harmonic functions and Poisson integrals and then move into a detailed study of Hardy spaces. The classical Dirichlet problem is a running theme throughout the book and a variety of methods and techniques for the resolution of the problem ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) and Hilbert space methods are presented.